What is ln((x^5)/(sqrt(5)(1+x)^4)) ?

The solution to ln((x^5)/(sqrt(5)(1+x)^4)) is 5ln(x)-1/2 ln(5)-4ln(1+x)

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simplify ln((x^5)/(sqrt(5)(1+x)^4))

simplify

ln (\frac{x ^{5}}{5 ( 1 + x ) ^{4}})

5 ln (x) - \frac{1}{2} ln (5) - 4 ln (1 + x)

Solution steps

ln (\frac{x ^{5}}{5 ( 1 + x ) ^{4}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{x ^{5}}{5 ( 1 + x ) ^{4}}) = ln (x^{5}) - ln (5 (1 + x)^{4})

= ln (x^{5}) - ln (5 (x + 1)^{4})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (x^{5}) = 5 ln (x)

= 5 ln (x) - ln (5 (1 + x)^{4})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (5 (1 + x)^{4}) = ln (5) + ln ((1 + x)^{4})

= 5 ln (x) - (ln ((x + 1)^{4}) + ln (5))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((1 + x)^{4}) = 4 ln (1 + x)

= 5 ln (x) - (ln (5) + 4 ln (1 + x))

ln (5) = \frac{1}{2} ln (5)

= 5 ln (x) - (4 ln (x + 1) + \frac{1}{2} ln (5))

- (\frac{1}{2} ln (5) + 4 ln (1 + x)) : - \frac{1}{2} ln (5) - 4 ln (1 + x)

= 5 ln (x) - \frac{1}{2} ln (5) - 4 ln (1 + x)

lo g_{x} (\frac{1}{8})

ln ((\frac{2}{3} x) \frac{1}{2} y)

- 9 ln (t) + ln (6) - ln (6 + 1)

\frac{1}{3} lo g_{10} (x) - lo g_{10} (y) - 2 lo g_{10} (z)

lo g_{2} (\frac{1}{4}) - lo g_{2} (2 - x)

What is ln((x^5)/(sqrt(5)(1+x)^4)) ?
The solution to ln((x^5)/(sqrt(5)(1+x)^4)) is 5ln(x)-1/2 ln(5)-4ln(1+x)